I'm a graduate student in Math. But I never learnt Topology during my undergraduate study. Next semester, I am going to take Differential Geometry. I assume this course would require a background of Topology. So I would like to take advantage of this summer and learn some topology myself.

I don't need to become an expert in Topology. All I need is that after this summer, my topology knowledge will be enough for my Differential Geometry course.

So can somebody please recommend me a textbook? I'd be really grateful!

I assume you've done some cursory research on common topology texts already. Do you have any specific questions about the plethora of advice already available on the internet?
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Antonio VargasMay 28 '12 at 0:04

I would recommend $\textit{Topology}$ by Munkres. I am not at all interested in topology, but I would say it is my favorite math textbook. It is very well-written. I don't think you need much point set topology for differential geometry or algebraic topology. You probably just need to know about continuous functions, compactness, and connectedness.
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WilliamMay 28 '12 at 0:20

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You need very little general topology for differential geometry.
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André NicolasMay 28 '12 at 0:59

For pure point-set topology, Wilansky's book is impossible to beat.
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ncmathsadistMay 28 '12 at 0:24

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As you are the only person who answered, I will pick your answer. Thank you for all of you who commented as well.
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henryforever14May 28 '12 at 0:38

This is the book that my graduate course in point-set topology used. I found it to be a good book for someone who had never had topology (because I had not).
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GraphthMay 28 '12 at 1:26

Munkres is a classic for good reason,but Wilansky is indeed a great book for students already familiar with the elements of point-set topology from real analysis. We should all be very grateful to Dover for making it available again for a very low price.
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Mathemagician1234May 28 '12 at 2:28

Seebach and Steen's book Counterexamples in Topology is not a book you should try to learn topology from. But as a supplemental book, it is a lot of fun, and very useful. Munkres says in introduction of his book that he does not want to get bogged down in a lot of weird counterexamples, and indeed you don't want to get bogged down in them. But a lot of topology is about weird counterexamples. (What is the difference between connected and path-connected? What is the difference between compact, paracompact, and pseudocompact?) Browsing through Counterexamples in Topology will be enlightening, especially if you are using Munkres, who tries hard to avoid weird counterexamples.

These counterexamples can shed insight though. A great example involves showing that first countable and separable do not jointly imply second countable. This is achieved via the "bubble topology", an ingenious piece of mathematical craftsmanship.
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ncmathsadistMay 28 '12 at 18:56

I entered my graduate general topology course with no previous background in the field (save what I knew about the real line). Despite this, I had great success with Stephen Willard's General Topology.

+1.Willard is the Bible of point-set topology,the single most comprehensive text ever written on the subject. Again,Dover has done a huge service to mathematics students by making it available again in a cheap edition!
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Mathemagician1234May 28 '12 at 2:29

That's a fabulous book. I like it a lot.
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ncmathsadistMay 28 '12 at 1:29

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De gustibus: I much prefer Munkres to Kelley. Come to think of it, I also prefer Dugundji to Kelley. If one has the necessary maturity, Willard is perhaps a better choice than any of these.
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Brian M. ScottMay 28 '12 at 1:46